How to make folding Origami animation
I remember making an animation folding a crane in the past. 

It involved making a real crane, unfolding it then cutting a plane mesh where the folds are in the real one. You have to double up on each cut as real paper folds have a little thickness.

Then add an armature with a bone for each plane, joined at the doubled hinges, then animate the armature.  

I'll see if I can dig up the blend file and provide a clearer explanation. The thing I remember most it's that it is not trivial! Don't expect it to be done in an hour.